Question

I need help with number 4 i don’t know what to do

Answer 1

`log_(16)( 2x + 7 ) = ( ln(2x + 7) )/( ln(16) ) = ( ln(2x + 7) )/(2 ln(4) ) = (1)/(2) ( ( ln(2x + 7) )/( ln(4) )) = (1)/(2) log_(4)(2x + 7) = log_(4)( √(2x + 7) )`

Answer 2

See below for proof.

Explanation:

Given logarithmic equation:

`\log_(16)\left(2x+7\right)=\log_4√(2x+7)`

To show that the left side of the equation equals the right side, we can use logarithmic properties and simplify the expression.

Begin by changing the base to 4 by using the change of base formula:

`\boxed{\textsf{Change of base:} \quad \log_ba=(\log_ca)/(\log_cb)}`

In this case:

• 
  `a = (2x + 7)`
• 
  `b = 16`
• 
  `c = 4`

Therefore, applying the change of base formula to the left side of the equation gives:

`(\log_4(2x+7))/(\log_4 16)`

Now, rewrite 16 as 4²:

`(\log_4(2x+7))/(\log_4 4^2)`

`\textsf{Apply the log power law:} \quad \log_ab^n=n\log_ab`

`(\log_4(2x+7))/(2\log_4 4)`

`\textsf{Apply the log law:} \quad \log_aa=1`

`(\log_4(2x+7))/(2(1))`

`(\log_4(2x+7))/(2)`

`(1)/(2)\log_4(2x+7)`

`\textsf{Apply the log power law:} \quad n\log_ab=\log_ab^n`

`\log_4(2x+7)^{(1)/(2)}`

`\textsf{Finally, use the property: $a^{(1)/(2)} = √(a)$}`

`\log_4√(2x+7)`

Therefore, we have successfully shown that the left side of the equation is equal to the right side.